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1.1. Nearly Integrable Hamiltonian Systems. In this work we examine the system of Hamiltonian equations i = _ iJH , ~ = iJH iJcp iJl with the Hamiltonian function H = Ho(l) + eH. (I. cp). (1.1) where E: "1 is a small parameter, the perturbation E:Hl (I ,cp) is 2n- periodic in CP=CP1,"'CPS' and I is an s-dimensional vector, I = Il, *** I s The CPi are called angular variables, and the Ii action variables. A system with a Hamiltonian depending only on the action variables is said to be integrable, and a system with Hamiltonian (1.1) is said to be nearly integrable. The system (1.1) is also called a perturbation of the system with Hamiltonian Ho. The latter system is called un- perturbed. 1.2. An Exponential Estimate of the Time of Stability for the Action Variables. Let I(t), cp(t) be an arbitrary solution of the per- turbed system. We estimate the time interval during which the value I(t) differs slightly from the initial value: II(t)-I(O) I "1. The main result of the work is Theorem 4.4 (the main theorem) which is proved in [1]. This theorem asserts that the above-mentioned interval is estimated by a quantity which grows exponentially as the value of perturbation decreases linearly: 1/(t)-/(O)I 0 and b > 0 are given l.n Sec. 4 [IJ.